Quantum cascade lasers at 16 µm wavelength based on GaAs/AlGaAs

Quantum cascade lasers at 16 µm wavelength based on GaAs/AlGaAs Sara Anjum January 9, 2018 Advised by: Claire Gmachl This paper represents my own work in accordance with University regulations. /s/ Sara Anjum Abstract Mid-infrared (3-30 µm) quantum-cascade (QC) lasers are usually designed and realized in InGaAs/AlInAs on an InP substrate. InP is a commonly-used substrate for semiconductor lasers because of its compatibility with InGaAs and AlInAs for tuning the depth of quantum wells in quantum-cascade lasers and its utility as a waveguide cladding material. However, its two-phonon resonance energy corresponds almost exactly to the energy of 16 µm photons, and so makes InP-based 16-µm QC lasers inefficient and low performing. 16 µm lasers are important for BTEX or UF 6 sensors. GaAs is the next best substrate for such a laser. In this paper, we design a new 16-µm GaAs-based QC laser and provide the background understanding for QC lasers in general. The active and injector region period length is L p = 520.4 Å. The figure of merit for the differential gain coefficient is 1.32, which is low, but it exhibits carrier inversion and would lase, making it a good preliminary design upon which to make modifications to optimize gain and minimize losses. 1

Table of Contents I Introduction 3 I.1 History of Lasers.................................... 3 I.2 Motivation....................................... 3 I.3 Basic Physics of Quantum-Cascade Lasers...................... 3 I.3.1 Schrödinger s Equation............................ 4 I.3.2 Solving for the Wavefunction......................... 5 I.3.3 Eigenenergies.................................. 5 I.4 Active Region Designs................................. 6 I.4.1 Three-Quantum-Well Active Region..................... 8 I.5 Resonator and Waveguide Designs.......................... 9 I.5.1 Dielectric Waveguides............................. 10 I.5.2 Surface-Plasmon Waveguides......................... 11 II Substrate Challenges and Solutions 12 III Necessary Tools for Numerical Analysis 13 III.1 Material Parameter Values in Ternary Alloys.................... 13 III.2 Temperature Dependence of Bandgap........................ 13 III.3 Lattice Strain Effects on Bandgap.......................... 13 III.4 Effective Mass...................................... 14 III.5 Wavefunction Normalization.............................. 15 III.6 Carrier Lifetimes.................................... 15 IV Quantum-Cascade Laser Design 16 IV.1 Zero Applied Bias................................... 16 IV.1.1 Single Well................................... 16 IV.1.2 Two-Well System................................ 17 IV.1.3 Three-Well System............................... 19 IV.2 Biased Three-Well Active Region........................... 20 IV.3 Active + Injector Region Period........................... 21 V Conclusions and Outlook 22 VI References 23 2

I Introduction I.1 History of Lasers The first laser experimentally-realized laser was an optically-pumped ruby laser [1]. Such solidstate lasers, which are based on doped crystals or glasses, are more stable and have longer lives than their gaseous counterparts [1]. The first semiconductor laser was created in 1962 and made use of a p-n diode, which is made up of a semiconductor junction half doped with electron acceptor atoms and half doped with electron donor atoms [1]. Quantum-cascade (QC) lasers are semiconductor lasers that rely on electron tunneling through a series of quantum well structures and intersubband transitions for stimulated optical emission. Semiconductor diode lasers rely on interband transitions, in which electrons move between a semiconductor s conduction and valence bands, while QC lasers rely on intersubband transitions, which are transitions between different energy levels within the quantum wells in the conduction band of the semiconductor structure. QC lasers were first conceived in theory by R.F. Kazarinov and R.A. Suris in 1971 [2]. It wasn t until 1994 that QC lasers were first realized in practice, when Faist et. al. at Bell Laboratories created a QC laser that emitted at 4.2 µm [3]. This first QC laser made use of a three-well active region with a dielectric waveguide and a cleaved facet mirror cavity [3]. Since then, QC lasers have evolved in all sorts of varieties. There are several active region designs and a handful of waveguide and resonator designs [4]. I.2 Motivation Long-wavelength ( 16 µm) lasers have many applications, especially in trace-gas sensing of UF 6 for nuclear non-proliferation worldwide and BTEX (benzene, toluene, ethylbenzene, xylenes), which are carcinogenic industrial chemicals [1]. However, it is difficult to obtain laser emissions at such wavelengths because of the lack of semiconductors which can emit in this range at room temperature. One of the lowest-bandgap semiconductors is indium antimonide, whose bandgap of 0.17 ev corresponds to a maximum wavelength of 7.2 µm [5]. Lead-salt lasers have smaller bandgaps, but the materials lack long-term stability [6]. QC lasers have immense flexibility in emission wavelength because they rely on intersubband transitions, whose energies are determined by the widths of the quantum wells, rather than relying on interband transitions which depend on the material bandgap, like other semiconductor lasers [7]. This allows one to use technologically mature materials, such as indium phosphide (InP) and gallium arsenide (GaAs), and their alloys as the laser substrate and materials rather than small-band semiconductors which are sensitive to temperature and often difficult to process [3]. In addition, the use of a cascading quantum well structure allows a single injected electron to emit multiple photons [7]. This can permit large gain and yield high-power lasers. I.3 Basic Physics of Quantum-Cascade Lasers At the simplest level, a QC laser consists of a series of quantum wells in the conduction and valence bands of the semiconductor structure. The quantum wells have finite potential barriers and are under a bias voltage, which permits tunneling of free carriers through the wells and gives rise to a current. Because of the applied potential, electrons tunnel from the ground state of upstream wells to the excited states of downstream wells. When the electrons relax from these excited energy levels to the ground energy level of a given well, they emit photons. If we also introduce population inversion across an optical transition, so that there are more electrons in the higher energy levels within the quantum well than there are 3

in the lower energy levels, then we obtain optical gain, amplification, and lasing [4]. Figure 1 shows an energy band diagram of a quantum-cascade laser s active and injector regions: However, for net amplification and lasing to occur, the gain must be able to more than offset the losses that the light experiences. There are three major loss mechanisms [4]. The first arises from imperfect reflections off of the mirrors which form the laser s resonator, the region which picks out particular modes for laser emission [4]. The second arises from waveguide absorption, which occurs when electrons absorb light in the (doped) semiconductor regions and the metal contacts [4]. The third arises from resonant intersubband transitions in the quantum wells, which can occur if the energy transitions correspond to or are lower than the photons energies. Figure 1: Energy band diagram schematic of a QC laser [6]. The figure shows an electric potential applied to a structure of alternating quantum wells and barriers within the conduction band. Electrons emit photons in the active region. In the injector region, electrons tunnel through to the excited energy state of a downstream active region. This paper investigates potential QC laser designs which use GaAs/AlGaAs to emit light at 16 µm. It describes the basic physics governing different components of a QC laser, examines the challenges with using InP with InGaAs/AlInAs to make 16- µm lasers, and explains why GaAs paired with Al- GaAs is a superior alternative. It then presents an original active region design and describes the step-by-step process used to develop it. I.3.1 Schrödinger s Equation The following section presents a basic analysis of the electronic states for a single quantum well with finite potential barriers. Denoting the width of the well as d and the barrier height as qv 0, we can define the potential of the system as: V 0 d V (z) = 2 z d 2 0 z > d (1) 2 where the potential energy is given by qv (z) [8] [9]. Under normal laser operating conditions, an external applied bias potential would be added to Eq. (1). Because the conduction band states are coupled with the valence and other bands, the solution is more complex than the traditional one-band Schrödinger equation [9]. However, for bound conduction band solutions, which is of greatest interest because only bound electrons contribute to gain, the interactions can be reduced to an energy dependence of the effective carrier mass [9]. The effective mass becomes higher for higher-energy electrons [9]. Since now the effective mass is dependent on position and energy, using the momentum operator P z = i z we must write the kinetic energy operator as 1 T = P z 2m(z, E) P z = 2 2 z 1 m(z, E) z where z is the horizontal distance for this 1D well along the well width, E is the electron energy, itself a function of z, m(z, E) is the electron s effective mass, dependent on position and energy 4 (2)

inside the structure, and is the reduced Planck s constant [9]. This means that the Schrödinger equation for the the bound states in the quantum well system involved in QC lasers is a slight variant on the traditional form seen in simplified finite quantum well cases [9]: 2 2 z 1 ψ(z) + qv (z)ψ(z) = E(z)ψ(z) (3) m(z, E) z I.3.2 Solving for the Wavefunction The first term in Eq. (3), which represents T ψ(z), can be approximated by using a discrete derivative approximation df dz = f(z + δz) f(z δz) 2δz (4) to obtain [ 1 2δz 1 m (z + δz, E) ψ(z) z z+δz 1 m (z δz, E) ψ(z) z ] z δz = 2 ] [qv 2 (z) E ψ(z) (5) Now, applying Eq. (4) to expand all the first derivatives in Eq. (5) and transforming 2δz δz, we can solve for ψ(z + δz): ψ(z + δz) = {[ 2(δz) 2 [ ] 1 2 qv (z) E + m (z + δz/2, E) + 1 m (z δz/2, E) 1 m (z δz/2, E) ] ψ(z) } m (z + δz/2, E) (6) [ ] where m (z ± δz/2, E) 0.5 m (z, E) + m (z ± δz, E). Thus, knowing the wavefunction at a given point z 0 permits can iteratively solve for the wavefunction at all points after that. I.3.3 Eigenenergies The eigenenergies inside a well correspond to the values of E where both ψ(0) = 0 and ψ(z end ) = 0. z = 0 represents the edge of the left-most well, which is bound to the left by an infinite barrier, and z = z end represents the edge of the right-most quantum well, which is bound on the right by an infinite barrier [9]. Computationally, one can solve for the eigenenergies by iterating through possible values of E between 0 and the band offset energy, dividing up the well into segments of a finite length δz, and repeatedly solving for ψ(z + δz) to obtain the value of ψ(z end ) and see which values of E give ψ(z end ) = 0 [9]. 5

Figure 2: Determining the energy eigenstates in a quantum well numerically. The energy eigenstates in the quantum well on the let correspond to the energies where the wavefunction at the right edge of the well evaluates to zero [9]. I.4 Active Region Designs A laser relies on population inversion so that stimulated emission, which occurs when electrons interact with photons and recombine with holes to emit more photons, creates light amplification. However, population inversion relies on a larger number of carriers in the excited state than in the ground state at any given moment, and the very short lifetime of the upper state makes this configuration difficult to achieve [10]. A good active region design must be electrically stable under the applied bias voltage, achieve population inversion, and result in a high enough gain to compensate for losses such as waveguide losses [10]. The gain coefficient of a QC laser is given by [4] ( g = τ 3 1 τ ) 2 Γ 4πe2 z32 2 (7) τ 32 ɛ 0 n eff λ 0 2γ 32 L p where τ 32 represents the 3-2 non-radiative scattering lifetime for the laser transition [10] τ 3 is the effective lifetime of the n = 3 state and is given by 1 τ 3 = 1 τ 32 + 1 τ 31 + 1 τ esc (8) where τ 31 represents the scattering time for the 3-1 transition, and τ esc represents the tunneling time into the continuum [10] z 32 is the optical dipole matrix element, which indicates how good a transition is and whether it is likely to lase [4] γ 32 represents the full-width half maximum (FWHM) of the broadened laser transition; the broadening is the result of electron scattering and coupling with energy bands which form when multiple quantum wells are coupled [4] L p is the length of one active + one injector region [4] Γ represents the overlap factor, which describes the amount of overlap of the light with the active material of the waveguide [10] 6

n eff is the effective waveguide index of refraction [4] λ 0 is the wavelength in vacuum [4] For the purposes of this paper, which is primarily concerned with active region design, and based on the information provided by the software used for this work, optimizing gain will involve maximizing z 32 because achieving gain requires that the desired transition from n = 3 to n = 2 occur often. The scattering lifetimes from the n = 3 state, τ 32 and τ 3, must also be maximized since excitation of carriers into higher energy states is just as likely as relaxation of carriers into lower energy states. Thus, achieving gain, which requires more emission than absorption, demands that carriers in excited energy states survive longer than carriers in the ground state, so τ 3 and τ 32 must be made as large as possible. It will also involve minimizing the scattering lifetime from the n = 2 state, τ 2, to keep the ground states as sparsely populated as possible, so that there is less Figure 3: Schematic of population inversion in a laser [11]. In (a), there are more carriers in the ground state because few carriers gain the necessary thermal energy to be excited into higher states. In (b), external intervention excites carriers into higher energy states. In QC lasers, this is accomplished by the tunneling of carriers in electrically biased wells. photon absorption than emission. This can be accomplished via the emptying of ground states quickly through the longitudinal optical (LO) phonon interaction. LO phonons out-of-phase vibrations of atoms in a unit cell. Reducing photon absorption to keep gain high also requires keeping the active and injector regions as short as possible by minimizing L p. Since the active region lengths will be fixed to a large extent by the desired wavelength of light, the important thing is to keep the injector region short. In thinking about the design of the gain region, it is important to realize that the energy transition from the first excited energy level (n = 2) to the ground energy level (n = 1) in mid-infrared lasers often corresponds to the energy of LO phonons. Electron scattering via such phonons occurs on shorter timescales than electron relaxation to emit photons; QC lasers take advantage of this to create population inversion. Electrons tunnel quickly through barriers from higher-energy well states to the excited n = 3 states of the next well before relaxing down to the n = 2 energy state. From there, they are typically scattered and relax down to the n = 1 state via an LO phonon before tunneling through to the next well. Because the LO phonon scattering timescale is much shorter (on the order of a fraction of a picosecond) than the electron relaxation timescale from n = 3 to n = 2 (a couple of picoseconds), the n = 2 level empties faster than the n = 3 level does. In addition, the fast tunneling time from the higher-energy well s n = 1 state to the next well s n = 3 energy state also contributes to a larger population in the n = 3 state than in the lower energy state. This allows population inversion to exist in a QC laser. For instance, in the first QC laser experimentally created, the tunneling time through the barriers was less than 0.5 ps and the phonon scattering time τ 2 was around 0.6 ps, but the electron relaxation time from n = 3 to n = 2, τ 32, was around 4.3 ps [3]. The variations on active region designs involve different ways of optimizing population inversion as well as keeping waveguide losses low and transition linewidths narrow, since design requirements to achieve these goals can vary as properties like lasing wavelength change [10]. In addition, different designs are more or less robust to temperature changes; thus, it is essential to pick a design which is most resilient to temperature increases so that it can be operated as close to room temperature as possible. This is particularly true for long-wavelength lasers, which have increased phonon populations and lower phonon-emission-induced electron scattering times [4]. Higher temperature operation also introduces the thermal excitation of electrons into the higher 7

energy level, which broadens the laser spectrum and both introduces other wavelengths besides the desired emitting wavelength and reduces laser gain [4]. Offsetting these effects requires increasing the gain by, for instance, designing a structure with longer carrier lifetimes in the n = 3 energy level and using more active region and injector stages for more cascading action [4]. Adding more stages will also reduce the threshold current density, which will reduce the issue of laser heating and the resulting thermal spread of electrons which reduces gain [4]. I.4.1 Three-Quantum-Well Active Region The three-quantum-well active region design was the first one to be used for QC lasers [10]. Such a design could either rely on vertical electron transitions, where the electron wavefunctions of the n = 3 and n = 2 state are located in the same spatial location, or on diagonal transitions, where the n = 3 and n = 2 wavefunctions are in slightly different locations in space [4]. In both vertical transition and diagonal transition designs, there are two main quantum wells within the active region, of which the second is thinner so that its energy levels are higher and can be brought into resonance with the first well s energy levels [10]. When these wells are brought together and a certain potential bias is applied, the ground states of the two wells are aligned and brought into resonance, and the electric field splits this resonant ground state into two energy levels, n = 1 and n = 2 [10]. The n = 3 state, then, is close to the first excited energy level in the first of these two quantum wells. Both designs also consist of a thin quantum well which precedes the two main, thicker quantum wells [10]. This reduces electron leakage from the injector directly into the n = 2 or n = 1 energy states because the thinner well yields higher energy states for all n, so it becomes more likely that the electrons will tunnel through to the n = 3 state [4]. The three-well active regions are separated by doped injector regions as shown in Figure 1. One advantage of the injector regions is that they give rise to minibands and minigaps because the quantum well barriers within them are thin, yielding wavefunctions which interact across quantum wells and creating closely overlapping energy levels and forbidden regions where wavefunctions cannot propagate. An elementary calculation of tunneling probability p T as shown in Griffiths [8], which is proportional to the square of the amplitude of the tunneling wavefunction, shows that p 1 T 1 + V0 2 ( ) b 4 E (V 0 E ) sinh2 2m E (9) where p T is the tunneling probability, E is energy of the electron in the barrier, b is the barrier width, and V 0 is the energy offset between the bottom of the well and the barrier [8]. E is the electron energy as measured with respect to the barrier energy of V (z) = 0, as indicated in equation (1), so E < 0. Since a higher energy state, like n = 2, is closer to the barrier potential than a lower-energy state, the tunneling probability is higher, and therefore the escape time is shorter [10]. However, when the n = 2 level is centered at a minigap of the injector region that follows, the wavefunction cannot propagate into this gap, so this makes the carrier lifetime in the n = 2 state longer than that in the n = 1 state [10]. A similar principle can be applied to higher energy levels. Injector region minibands are designed to be flat under bias in a way such that the level 1 of the preceding active region is in resonance with the injector region minibands, which is in resonance with level 3 of the following active region [4]. In addition, the energy separation between minibands is much lower than the energy of an LO phonon [4]. These two characteristics allows for carrier tunneling between these resonant layers without much relaxation in the injector region [4]. 8

Diagonal Transition Active Region In a diagonal transition active region, the thin first well of the active region is made just thin enough so that its ground state energy level aligns with the first excited state of the neighboring quantum well [10]. When this happens under bias, the electric field splits the resonant energy level into two, creating the n = 3 and n = 4 energy levels [10]. Since the wavefunction is resonant across the barrier between these wells, the transition from the n = 3 to the n = 2 state is diagonal since the electron can transition fairly easily from the n = 3 level in the thin well directly to the n = 2 level in the thicker well [10]. Figure 4 to the right illustrates the diagonal transition where the electron goes from the n = 3 state of the first, thin well to the n = 2 state of the well. Because the oscillator strength, which is related to the square of the optical dipole matrix element of a transition, is lower in the diagonal transition, the transition is not as strong, which results in a lower gain [10]. However, the lifetime τ 3 in the n = 3 energy states as well as τ 32 are longer, which allows for more inversion. Thus, depending on the parameters of the structure the longer lifetimes can allow a higher gain if they compensate the smaller z 32 [10]. Vertical Transition Active Region Figure 4: Schematic of a diagonal transition active region [12]. The photon-emitting transition occurs from the n = 3 state of an upstream well to the n = 2 state of a downstream well. Vertical transitions occur when the first, thin well in the active region is thin enough such that its ground energy state is of higher energy than the n = 3 energy state of the next well down. This way, the transition is vertical because the electron must relax after tunneling through to the next well before dropping down to the n = 2 energy state. Figure 5 illustrates the vertical transition. Because vertical transitions require high spatial overlap between the wavefunctions of the active region wells, the optical dipole matrix element z 32 of the n = 3 to n = 2 transition is high, so the transition is strong and contributes to increasing gain as seen in Equation (13). This means that the laser can withstand a large range of electric potential biases and indicates that it can function well over a large temperature range, although it still requires at least 6 stages of active regions and injectors for any sort of room-temperature lasing action at all [4]. However, this strong spatial overlap also reduces τ 32, which contributes to lowering gain [10]. In the design presented in this paper, a vertical transition active region was chosen because of the higher temperature resilience and the ease of manipulating vertical transitions. I.5 Resonator and Waveguide Designs Waveguides and resonators, which confine and amplify light, distinguish lasers from LEDs because they prevent the light from escaping before it can be amplified [4]. However, as 9 Figure 5: Schematic of a vertical transition active region [12]. The photon-emitting transition occurs from the n = 3 state of a given well to the n = 2 state of the same well.

light travels through the waveguide, it experiences losses because the same material that produces the light can also absorb it, which reduces the number of photons and therefore the intensity of the emitted light [4]. Thus, reducing waveguide losses is essential to making the most of the achievable laser gain. For nearinfrared wavelength ranges, dielectric waveguides are the obvious choice however, the choice is more complicated for longer wavelengths, which includes 16 µm lasers [4]. I.5.1 Dielectric Waveguides Indium gallium arsenide (InGaAs) and indium aluminum arsenide (InAlAs) can be grown on an InP substrate to create a mid-infrared QC laser. The index of refraction is smaller for InP, a potential substrate material [13], than it is for InGaAs and AlInAs, which are materials used in active regions [14]. Because of that and the close lattice constant matching between these semiconductors, InP can be used as a substrate to surround the active regions and create a waveguide. A waveguide is created with a central region of relatively high index of refraction and a surrounding region of lower index of refraction, which yields total internal reflection for light traveling laterally and confines the light. Figure 6 illustrates the total internal reflection and the resulting confinement inside a dielectric waveguide. In order to achieve as low a threshold current as possible by confining as much of the light as possible, the difference between the refractive index of the core and the cladding layer should be as high as possible so that there is as high an overlap between the guided optical mode and the active and injector regions as possible [4]. The effective refractive index of the core can be increased by sandwiching the active region in between layers of the active-region material with the Figure 6: Schematic of a dielectric waveguide [15]. Light travels out from the source in all directions, but light which comes in at an small angle (relative to the line perpendicular to the cladding plane) will be reflected back into the waveguide. This internal reflection keeps the light confined within the waveguide. higher index of refraction [4]. For instance, in the case of an active region made of In- GaAs and InAlAs, the active region can be sandwiched in between layers of InGaAs before being surrounded by the InP substrate [4]. In considering a dielectric waveguide and the potential materials for a QC laser, one must remember that typically a portion of the waveguide layers is doped [4]. The free carrier absorption and the onset of the plasma edge, which is where the resonant motion of electrons in a semiconductor makes it look like a metal, reduce the index of refraction by a few percent [4]. While the dielectric waveguide works for QC lasers which use InP as the substrate, they are less suitable for a GaAs substrate-based laser. This is because GaAs has a higher index of refraction than AlGaAs and thus cannot serve as a cladding layer with which to surround the active and injector regions for total internal reflection to occur and confine the light [4]. In addition, if one tried to make a core out of a combination of AlGaAs and GaAs to reduce the core s index of refraction and use AlGaAs as the cladding, then the high AlAs fraction required for the fabrication of the core would be impractical to actually dope for effective current transport [4]. 10

Thus, a plasmon-enhanced dielectric waveguide would be a better design, based on previous demonstrated success [4]. I.5.2 Surface-Plasmon Waveguides In a surface plasmon waveguide design, the electromagnetic wave modes are surface waves propagating at a boundary between a metal and a semiconductor [4]. A plasmon is an oscillation of a free electron in a metal [16]. An electron attracted to a positive nucleus inside a material can be modeled as a particle attached to a spring [17]. This is true because if the potential energy resulting from any binding force is Taylor expanded around equilibrium, then the quadratic term is the first nonzero term with dynamic significance when a particle is displaced from equilibrium [17]. When there is an electromagnetic wave in the vicinity of this spring, it is subject to a driving force qe 0 cos(ωt) where E 0 is the wave s amplitude at the point where the electron is located and ω is the electromagnetic wave frequency [17]. The system can be represented by the spring system equation m d2 x dx + mγ dt2 dt + mω2 0x = qe 0 cos(ωt) (10) where γ is the damping constant used to quantify the damping of the electron oscillation and ω 0 quantifies its natural oscillation frequency [17]. Dividing both sides and solving the differential equation yields q/m x(t) = ω0 2 ω2 iγω E 0e iωt (11) The denominator consists of a complex term, which indicates that the electron oscillation is out of phase with respect to the incoming electromagnetic wave. The denominator gives a phase difference of ( ) γω φ(ω 0, ω) = tan 1 ω0 2 (12) ω2 When ω ω 0, which is the case in metals, then φ π, and the oscillations are completely out of phase with the incoming electromagnetic wave [17]. The Drude model of solids assumes that electrons are free particles inside a metal while the nuclei are immobile [18]. Because the electrons are not bound to the nuclei, the spring constant of the bond mω0 2 = 0. Given that the frequency of visible light is on the order of 10 14 Hz, ω ω 0 and φ π, so the free electron oscillations in a metal are out-of-phase with respect to the incoming electromagnetic wave [19]. The out-of-phase oscillations, as well as oscillations in surface charge density, cause high reflectivity of light and allows light to remain confined to the surface of a metal-semiconductor interface [16]. In addition, in a metal following Ohm s Law, once the free charge has dissipated and there is equilibrium, we have the following for Maxwell s equations: E = 0 B = 0 E = B t B = µɛ E t + µσe (13) where σ is the conductivity of the metal and ɛ is the permittivity of the metal [17]. Applying the curl yields 2 E = µɛ 2 E E t 2 + µσ t, 2 B = µɛ 2 B B t 2 + µσ (14) t which yield plane-wave solutions with complex wave numbers [17]: Ẽ(z, t) = Ẽ0e i( kz ωt), B(z, t) = B0 e i( kz ωt) (15) 11

where k is the complex number k 2 = µɛω 2 + iµσω (16) Because of this complex wave number, an electromagnetic wave propagating through a medium exponentially decays into the metal and semiconductor it travels [17]. Figure 7 demonstrates this decay of the magnetic field. Only the component of the electric field propagating in the direction of material growth in a QC laser gives rise to a nonzero dipole matrix element [10]. This means that transverse-electric (TE) polarized light cannot propagate through the laser. Thus, the light propagating through the laser structure must be transverse-magnetic (TM) polarized. While the absorption of TE can be nonzero in practice, it s typically only a small fraction of the absorption of TM-polarized light [10]. For the TM modes, a particular bounded mode is allowed when the dielectric constant changes sign at an interface between two materials [10]. This happens at the boundary between a metal and a semiconductor, as shown in Figure 7. Figure 7: Graph of the transverse magnetic component of the electromagnetic modes propagating through the waveguide [20]. The propagation direction is taken to be x. The graph shows that the field component is strongest near the surface, illustrating the confinement. The high reflectivity and decay of light s amplitude as it propagates through the metal prevents much light from escaping the surface of the metal-semiconductor interface, just as total internal reflection confines light within a waveguide [4]. However, taking the square root of Eq. (22) gives us a complex wavenumber of the form k = k+iκ where κ ω and tells us the extent of the decay of the wave where a higher κ means the electromagnetic wave can penetrate less into the metal [17]. Since κ is smaller for smaller frequencies of light, a metal-semiconductor waveguide tends to be less effective for mid-infrared wavelengths, but is advantageous for longer wavelengths [4]. However, of the metals used, gold is the best for light with wavelength λ 15 µm [4]. II Substrate Challenges and Solutions The first QC laser was demonstrated on an InP substrate, and typically InP is preferred over GaAs as a substrate because it yields better device performance and because of a range of other device properties [21]. Such properties include the higher bandgap offset between InGaAs and AlInAs, which can be lattice matched to InP, than between GaAs and AlGaAs [21]. However, GaAs is successfully produced on an industrial scale, as compared to InP, and Al x Ga 1 x As is lattice-matched to GaAs for all values of x between 0 and 1 [21]. This means that any bandgap offset up to that between AlAs and GaAs offset can be chosen from 0 to 500 mev [6]. In addition, GaAs is better suited than InP as a substrate for a 16-µ m QC lasers specifically. InP exhibits a two-phonon resonance at an energy of 626 cm 1, while a 16-µ photon corresponds to a wavenumber of 625 cm 1 [22]. This means that the 16 µm photons are very prone to absorption, and would not propagate well through the material at all. Even though GaAs would be a better substrate for a 16-µm laser for these reasons, it still has several material defects relative to InP that it needs to overcome in order to be a good laser. First, the smaller band-gap offset between the different active region materials means that extra effort must be put in to prevent electron escape from an excited energy state in a quantum well 12

to the continuum, which make the escaped electrons unavailable for lasing [21]. In addition, since its electrons have a higher effective mass than InGaAs electrons, which are used with InP, scattering lifetimes are lower, which reduces population inversion [21]. Focusing on design improvements to offset these material drawbacks, such as good active regions and waveguide designs, can allow one to make a good GaAs QC laser [21]. This paper will focus on active region design. III Necessary Tools for Numerical Analysis Section I.3 of the paper discussed Schrödinger s equation for bound states inside a QC laser s well, and explained how to solve for the wavefunction and eigenenergies within the system. An open-source software called ErwinJr, developed in our research group, which simulates devices like QC lasers, was used to design the active region. The following sections will elaborate on some of the other equations used in the code to calculate wavefunctions and eigenenergies. III.1 Material Parameter Values in Ternary Alloys For one, the active regions involve ternary alloys which can vary in composition. For any material parameter P, such as the permittivity of the medium, the value of P (A x B 1 x C) can be defined in terms of the parameter values of the semiconductors AC and BC: P (A x B 1 x C) = xp (AC) + (1 x)p (BC) + x(1 x)c B (17) where C B is a bowing parameter which quantizes the parabolic contribution to the alloy s parameter [9]. III.2 Temperature Dependence of Bandgap The bandgap of different materials is affected by temperature. The Varshni formula, an empirical formula, provides the bandgap at temperature T with respect to the temperature at T = 0 K: E g (T ) = E g (0) αt 2 (18) T + β where E g (T ) is the bandgap at a given temperature T, α and β are parameters which are experimentally determined for different materials [9]. With the bandgap, the barrier height of quantum wells depends on temperature. III.3 Lattice Strain Effects on Bandgap Lattice strain is an inevitable result of using ternary semiconductor alloys because the lattice constant changes with composition, which means that altering the composition of an alloy to change the band gap will introduce strain [9]. However, strain is undesirable as it can cause defects such as cracking at the interface [23]. Most III-V alloys have a zincblende crystal structure, so they acquire biaxial strain when grown on a substrate of different lattice constant [9]. Thus, only the diagonal elements of the 3D tensor are nonzero [9]. Labeling the axis of 13

growth as z, we can calculate the values of strain ε xx, ε yy, ε zz : ε xx = ε yy = a 0 a lc a lc ε zz = 2c (19) 12 ε xx c 11 where a 0 is the substrate lattice constant, a lc is the lattice constant of the layer that s being grown on (epitaxial layer), and c 11 and c 12 are the materials elastic stiffness constants. The change in band gap results from the relative change in volume due to the strain: δv δe Ec = a c V, δe δv Ev = a v V where δe Ec is the shift in conduction band energy and δe Ev is the shift in valence band energy [9]. a c and a v in (20) are the conduction and valence band hydrostatic deformation potentials, respectively [9]. δv is the change in volume, and V was the total original volume [9]. The band gap shift is found by just adding δe Ec + δe Ev [9]. (20) III.4 Effective Mass The effective mass can be approximated as a function of position z and energy E by using the k p perturbation theory [9]. In a periodic structure, Bloch s theorem says that the solution will take the form Φ nk = u nk (r)e i(k r), which when plugged into the Schrödinger equation, yields ( p 2 2m + k p m + 2 k 2 2m + V ) u nk = E nk u nk (21) where the term in parentheses is the Hamiltonian of the system, in the form H = H 0 + H [24]. Here, H 0 p2 2m + V, H k p m + 2 k 2 (22) 2m In the nondegenerate case, which is what we are concerned with for a single well, perturbation theory tells us that u nk = u n0 + un0 k p u n 0 u n m E n0 E 0 n 0 n n E nk = E n0 + 2 k 2 2m + 2 m 2 n n u n0 k p u n 0 2 E n0 E n 0 E n0 + 2 k 2 2m where u n0 is the term when k = 0 and n and n represent distinct wavefunction solutions to the equation corresponding to different energy values E n0 and E n 0 [24]. m is the electron mass in vacuum [24]. The final approximation is good for small values of k [24]. Comparing the left and right-hand sides of the equation gives us the following expression for effective mass [24]: 1 m = 1 m + 2 m 2 k 2 n n u n0 k p u n 0 2 E n0 E n 0 (23) (24) A more general analysis of this sort yields the effective mass expression of { [ ]} 1 m (z, E) = 1 1 + 2F + E P 2 1 m 3 E + EC LH Γ + E + EC SO Γ (25) 14

where E Γ C LH is the bandgap along the light hole valence band dimension and E C SO is the bandgap along the split off valence band dimension, while E P is the interband matrix element between the s-like conduction bands and the p-like valence bands [9] [25]. Meanwhile, F is the Kane parameter representing the second-order k p term. Since there are three dimensions for the wavenumber k, the holes don t have the same effective mass in all directions which gives rise to light holes and heavy holes. In addition, there is a split off band just below the valence band. III.5 Wavefunction Normalization In order for the wavefunction to adequately reflect probability densities and energies, they must be properly normalized. The normalization condition is given by ψc ψ c + ψlh ψ LH + ψso ψ SO = 1 (26) where ψ c is the conduction band wavefunction, ψ LH the light-hole wavefunction, and ψ SO the split-off wavefunction [9]. As Franz shows in his thesis [9], this can be written entirely in terms of ψ c by the following: E ψ c + 1 E + E v ψ c = 1 (27) where E is the electron energy and E v = 2E LH + E SO 3 is the average valence band energy [9]. III.6 Carrier Lifetimes Finally, one of the most important mechanisms for lasing is the manipulation of carrier lifetimes. According to Fermi s golden rule, the transition rate from one state to another is given by R = 1 τ = π 2 i Hint f 2 ρ(e f E i ω) (28) where i represents the wavefunction of the initial state, f the wavefunction of the final state, τ the carrier lifetime in the initial state, H int the interaction Hamiltonian, and ρ the energy density of final states [10]. The total Hamiltonian for an electron in a conduction band, which includes the light-matter interaction, is and yields the interaction Hamiltonian H = ( P qa) 2 2m + qv c ( r) (29) H int = q m (A P ) (30) which can be used to determine the transition rate [9]. By using the appropriate expressions for a QC laser system and working out all the necessary 3D integrals, we can obtain the transition rates W spon = 1 = q2 n eff ω0 3 τ spon 3 ɛ 0 c 3 (31) 0 for spontaneous emission and W stim = 1 = πq2 zul 2 N k,ϑ L(ν) (32) τ stim hv k ɛ 15

for stimulated emission, where N k,ϑ is the number of mode photons, z ul is the optical dipole matrix element given by z ul ψc z ψ l c u, Eu E l = ω 0 defines the frequency of the transition, V k is the mode volume, and δν/(2π) L(ν) = (ν 0 ν) 2 + ( ) δν 2 2 describes a Lorentzian lineshape function which expresses the broadening that occurs such that the transition frequency ω 0 is not a single frequency [9]. The phonon lifetime can be calculated in a similar way. The Frohlich interaction is the interaction of charge carriers with the polarization resulting from the relative displacement of negative and positive charges in a polar semiconductor [9]. The corresponding Frohlich Hamiltonian is H F = 2π ω LO q 2 e iq r (ɛ ɛ s )V Q Q 2 a Q (33) Q where ω LO is the phonon energy, ɛ is the high frequency permittivity, ɛ s is the static relative permittivity, V Q is the volume of the system, and Q is the magnitude of the phonon wavevector Q [9]. The lifetime can be calculated using Fermi s golden rule just as above, and its dependence on temperature can be expressed using the Bose-Einstein occupation distribution: 1 τ ul (T ) = 1 τ ul (0) ( 1 + 2 e ω LO/(k B T ) 1 While there are other figures and formulas involved, the above discussion covers the most important properties and parameters that will be manipulated for the laser active region design. ) (34) IV IV.1 Quantum-Cascade Laser Design Zero Applied Bias A ground-up approach was used to design the quantum well structure for a 16-µm GaAs QC laser. 16 µm corresponds to an energy of E = hc λ = hc 16 10 6 77.7 mev (35) m The first step was to begin with a single well and determine the appropriate well width for a 77.7 mev transition between the n = 2 state and the n = 1 state. These states were used because coupling more wells will cause these energy levels to split and form bands, so for a single well the first two energy levels are the relevant ones. Then, additional wells were coupled to the first one. Well and barrier widths were adjusted to optimize the dipole matrix element while maintaining the appropriate energy level differences for both the optical and LO phonon transitions. Once this was accomplished, a bias voltage was added, and well and barrier widths were further adjusted. Finally, an injector region was created to carry the electrons from one active region to another. This section will describe the results at each stage. IV.1.1 Single Well The following analysis of a single, unbiased quantum well with finite barriers entails the simplifying assumption that the electron s effective mass was independent of position or electron energy. 16

To the left is a graph of the potentials and eigenstates of a single quantum well made from GaAs/Al.195 Ga.805 As. According to Griffiths analysis [8], the conditions of continuity of wave functions and their derivatives across barrier boundaries yields the condition { (z0 ) 2 tan(z) Even solutions 1 = z cot(z) Odd solutions (36) Figure 8: Energy band diagram of one quantum well made of GaAs/Al.195 Ga.805 As. The band offset is 189 mev. This composition yields an energy transition of 77.5 mev (16.0 µm), a dipole matrix element of 25.6 Å, and an LO scattering time of 0.56 ps. where z = d 2m(E + qv 0 ) and z 0 = d 2mqV 0 2 2 when d is the width of the quantum well as given in Eq. (1) [8]. Thus, adjusting z 0, which encompasses both the well width and well height V 0, allows one to solve for the eigenenergies [8]. Solving this equation to get the desired well width gives d 111 Å, which gives a transition energy of 77.5 mev between the n = 1 and n = 2 states. This corresponds to a wavelength of 16.0 µm. Both the well and the barrier are made of Al x Ga 1 x As, so different values of x are necessary for the well and the barrier. Thus, the lattice constants of the two regions will be different, and some strain is inevitable. My choice of x = 0.195 for the barrier keeps the net strain at 0.008 % while yielding a band offset of 189 mev, which is enough to confine the n = 1 and n = 2 wavefunctions. A well width of d = 101.8 Å, corresponding to 36 atomic monolayers (ML), gives an energy difference of E 2 E 1 = 77.5 mev. This corresponds to a wavelength of 16.0 µm. Evidently the simplified assumption of a constant effective mass works quite well as a first approximation, giving an answer within 10% of the actual well width. IV.1.2 Two-Well System A second quantum well was coupled to the first, with both wells having the same width, so that the eigenenergies would align. The potential of the wells can be described as 0 z > b V (z) = 2 + d, z < b 2 b V 0 2 < z < b (37) 2 + d where b represents the width of the barrier coupling the two quantum wells. Coupling two wells together splits the energy levels so that the energy level n = 1 in the single well has now split into a band with wavefunctions corresponding to n = 1 and n = 2, while the energy n = 2 in the single well has now split into a doublet with wavefunctions corresponding to n = 3 and n = 4. We now have a ground band made up of n = 1 and n = 2 and an excited band made up of n = 3 and n = 4. Thus, now the energy difference between n = 2 and n = 3 is relevant for the optical transition, while the energy difference between n = 1 and n = 2 is relevant for the phonon transition. 17

Figure 9 (right) shows a diagram of a sample twowell system. Table 1 below shows the values of the variables which were tuned and resulting values of system parameters. In Table 1, d represents the well widths of both quantum wells, b gives the coupling barrier width, x gives the fraction of aluminum in the barriers, and E c gives the bandgap offset between the bottom of the well and the barrier. E 3 E 2 = 77.5 mev, which corresponds to a wavelength of 16.0 µm. Both the dipole matrix element and the LO phonon scattering time for this transition have increased relative to the one-well case. However, the E 2 E 1 = 25.3 mev, which is much lower than the phonon energy range of 35.3-38.8 mev. This is problematic because achieving carrier inversion for gain requires the energy difference between n = 1 and n = 2 to correspond to the energy of a single phonon so that the scattering time is smaller than that of the 3-2 transition. Attempting to increase the 2-1 transition energy would require making the barriers thinner so that Figure 9: Schematic of a two-well system made of GaAs/Al.24 Ga.76 As coupled by an 8.5 Å thick barrier. The bandgap offset is 223 mev, and the width of each well is 79.1 Å. The 3-2 transition energy is 77.5 mev (16.0 µm), the dipole matrix element of that transition is 34.0 Å, and the LO phonon scattering time is 0.95 ps. The 2-1 transition energy offset is 25.3 mev, so the LO scattering time is many orders of magnitude larger. the coupled wavefunction bands broadened. This would lower the energy of the n = 3 eigenstate while increasing the energy of the n = 2 eigenstate relative to the n = 1 eigenstate. However, it is impractical to fabricate layers thinner than three atomic MLs, so this two-well design would be unfeasible in practice. Parameter Value d 79.1 Å b 8.5 Å x 0.24 Lattice strain 0.008 % E c 223 mev 34.0 Å z 32 3-2 transition scattering time E 3 E 2 E 2 E 1 0.95 ps 77.5 mev 25.3 mev Table 1: Table of the quantities manipulated and resulting parameter values in a two quantum-well system. 18

IV.1.3 Three-Well System Just as coupling two wells together split the energy levels into bands, each of which contained two distinct energy states, coupling three wells together split the energy levels into bands containing three distinct energy states each. However, all six energy states would not fit inside the well because of the barrier height. Figure 10 (right) shows the schematic of an unbiased three-well system. Table 2 below shows the values of important system parameters: Parameter Value d 67.8 Å b 14.1 Å x 0.188 Lattice strain 0.007 % E c 184 mev 38.8 Å z 32 3-2 transition scattering time 1.5 ps 2-1 transition scattering time E 3 E 2 E 2 E 1 0.29 ps 77.7 mev 35.8 mev Table 2: Table of the quantities manipulated and resulting parameter values in a three quantum-well system. Figure 10: Schematic of a three-well system made of GaAs/Al.188 Ga.812 As coupled by 14.1 Å (5 atomic ML) thick barriers. The bandgap offset is 184 mev, and the width of each well is 67.8 Å (24 atomic MLs). The 3-2 transition energy is 77.7 mev (16.0 µm), the dipole matrix element of that transition is 38.8 Å, and the LO phonon scattering time is 1.5 ps. The 2-1 transition energy offset is 35.8 mev, and the LO scattering time is 0.29 ps. 19

The three-well system was able to achieve a 3-2 transition energy of 77.7 mev (16 µm) and a 2-1 transition energy of within the LO phonon energy range using GaAs/Al.188 Ga.812 As. This composition reduced the net strain marginally to 0.007 %. Both z 32 and the LO scattering time for the 3-2 transition are higher than those in the two-well case. In addition, the LO scattering time for the 2-1 transition is only 0.29 ps, which would result in good inversion if these figures were to remain when the system was biased. IV.2 Biased Three-Well Active Region Biasing the structure puts the first quantum well at at a higher potential than the last one and tilts all the well bottoms themselves. Thus, if all the widths are kept the same, the energy levels are misaligned, with the wells at lower potential having energy levels placed lower. To compensate for this, the wells at lower potentials are made thinner than the wells at higher potentials. If the structure was unbiased, the ground and excited states of the thinner wells would be at higher energies than the thicker wells. If tuned to appropriate thicknesses, the energy differences can compensate for the bias and make the energy bands straighter, reducing broadening and keeping the transitions strong. Below is the diagram of a modified three-well structure with a 30 kv/cm bias applied to it: When a bias of 30 kv/cm was applied, the upstream wells were wider than the downstream wells. In addition, the barriers were also of different thicknesses. Parameter Value d 1, d 2, d 3 84.8 Å, 67.8 Å, 50.9 Å b 1, b 2 11.3 Å, 19.8 Å x 0.22 Lattice strain 0.008 % E c 208 mev 36.1 Å z 32 3-2 transition scattering time 1.1 ps 2-1 transition scattering time E 3 E 2 E 2 E 1 0.45 ps 77.7 mev 37.3 mev Table 3: Table of the quantities manipulated and resulting parameter values in a 30 kv/cm-biased three quantum-well system. Figure 11: Schematic of a 3 quantum well system with a 30 kv/cm bias. GaAs/Al.22 Ga.78 As was used, resulting in a bandgap offset of 208 mev. The 1st well is 84.8 Å thick, the 2nd 67.8 Å thick, and the 3rd 50.9 Å thick. The 1st and 2nd wells have an 11.3 Å thick barrier, while the 2nd and 3rd wells have a 19.8 Å thick barrier between them. The 3-2 energy transition is 77.7 mev with a dipole matrix of 36.1 Å and an LO phonon scattering time of 1.1 ps. The 2-1 energy transition is 37.3 mev with an LO scattering time of 0.45 ps. 20